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I recently read Palle Yourgrau's book "A World Without Time" about Gödel's contribution to the nature of time in general relativity.

Gödel published his discovery of closed timelike curves in 1949. Many years later (in 1961), S. Chandrasekhar and James P. Wright pointed out in "The geodesic in Gödel's universe" that these curves are not geodesics, and hence Gödel's philosophical conclusions might be questionable. Again some years later, the philosopher Howard Stein pointed out that Gödel never claimed that these curves are geodesics, which Gödel confirmed immediately. Again much later other physicists have computed that these closed timelike curve must be so strongly accelerated that the energy for a particle with a finite rest mass required to run through such a curve is many times its rest mass. (I admit that I may have misunderstood/misinterpreted this last part.)

Questions

  1. This makes me wonder whether any particle (with finite rest mass) actually traveling on a closed timelike curve wouldn't violate the conservation of energy principle. (As pointed out in the comments, I made a hidden assumption here. I implicitly assumed that the particle traverses the closed timelike curve not only once or only a finite number of times, but "forever". I put "forever" in quotes, because the meaning of "forever" seems to depend on the notion of time.)
  2. I vaguely remember that light will always travel on a geodesic. Is this correct? Is this a special case of a principle that any particle in the absence of external forces (excluding gravity) will always travel on a geodesic?
  3. Is it possible for a particle to be susceptible to external forces and still have zero rest mass?
  4. Is it possible that Chandrasekhar and Wright were actually right in suggesting that Gödel's philosophical conclusions are questionable, and that they hit the nail on the head by focusing on the geodesics in the Gödel's universe?
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This makes me wonder whether any particle (with finite rest mass) actually traveling on a closed timelike curve wouldn't violate the conservation of energy principle. I don't think this follows at all from the fact that you need to input $E > mc^2$ to get a particle of mass $m$ around the CTC. You just need an external source of energy. –  Ben Crowell Aug 25 '13 at 22:16
    
@BenCrowell Regarding your comment, I had made the implicit assumption that a particle which traverses a closed timelike curve once is forced to do so forever. If it is possible for a particle to traverse a closed timelike curve only a finite number of times, then I agree that there is no need to worry about the conservation of energy principle. –  Thomas Klimpel Aug 25 '13 at 23:49
    
@ThomasKlimpell: CTC's are not elementary ideas. I have added a very crude analogy to give you a sense of the 'unphysical-ness' of the concept. Quite often in theoretical physics, ideas cannot be fully visualized with perfect detail. I urge you to read up the theory of causal structure. –  dj_mummy Aug 26 '13 at 7:11
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1 Answer 1

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I have not really studied Godel's metric, so I will only address questions 2 and 3 in a general metric (without specifically referring to Godel's metric).

Yes, light (in vacuum) will always travel on a null geodesics. Yes, particles remain on geodesics in absence of a net external force. Momentum means different things in the massive and mass-less case, since massive particles move on geodesics with timelike tangent vectors and mass-less move on null tangent vectors. 4-force is equal to the covariant derivative of 4-momentum along the tangent vector to its worldline. I will elaborate:

Let us assume a world in which quantum mechanics is bogus and all particles have a 'kick' (momentum) associated with them. A particle of light has a definite momentum associated with it. So its 'kick' can be redirected and/or diminished. Particles with mass also have this 'kick' and can also have it redirected and/or diminished. The 'kick' is redirected when 'kick' makes contact with the force applier i.e. they would be deviated from their geodesic motion.

Now, in particles with non-zero mass this kick is directly proportional to the 4-velocity. So applying a force on the particles changes its 4-velocity and deviates it from timelike geodesic motion.

However, for mass-less particles the 4-velocity does not exist (as proper time in their frame is 0). Applying force on the particle would also deviate it from null geodesic motion, but the tangent vector of it's motion would remain a null vector, so their net speed would still remain c in your local frame throughout the application of force.

Back to reality. In classical GR, we don't have any forces for these mass-less particles, but have forces (Electromagnetic forces) for massive particles. So we treat mass-less particles purely as waves with energy and momentum (that can't be changed by applying classical force). Note, in classical GR, in vacuum the speed of the EM wave can be reduced in dielectric media, but the fastest speed possible in the dielectric frame will still remain a null vector (speed of light).

In the above discussion, I treat gravitation as the structure of space-time and not as a force.

In Quantum Field Theory, observation is discontinuous and particles change in number and type between 2 successive measurements. There is a symmetry in these changes which leave net Energy and momentum invariant. So here force is irrelevant here and we treat photons as particles again.

Questions 1 and 4:

First of all there is no global conservation of energy in general relativity. There is only local conservation of energy. There are other methods used to get globally conserved quantities (like Killing vectors fields).

CTC's are looked upon as pathological entities. A whole lot of concepts in classical GR have to be revisited if if we accept CTC's in the acceptable causal structure of realistic spacetimes.

A lot of ideas we take for granted are thrown to the winds in such extreme spacetime. Let me give you a very crude and rough analogy:

-There is an astral chicken that lays an egg and dies, the egg hatches and the chick eats the egg and its parent, lays an egg, dies and so on..... Thus, the astral chicken's wordline is a CTC.

-Let's say you (moving along a normal geodesic) are at the event P (hatching of egg) and stay with the chicken till event Q (dying of chicken), the chicken will vanish suddenly after Q. Can you imagine the chicken vanishing?

-The egg also appeared suddenly in your past at P. Kind of like Marty in Back To the Future who appears and disappears suddenly. The egg-mass appears, turns into a chick and disappears, obviously from your viewpoint, energy is not conserved at all not even locally.

This is the best I can do without referring using math. Causal Structure is a very elementary theory, you will be able to understand it. This would help you better understand CTC's, which are not elementary at all. I recommend Wald's book on GR. In addition, here is a pdf by Thorne on some implications of CTCs. It is a moderately advanced paper, but very interesting.http://www.its.caltech.edu/~kip/scripts/ClosedTimelikeCurves-II121.pdf

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I'm confused by the "No" in "Yes, light will always travel on a geodesic. No, in classical GR, force only implies change in momentum." I initially interpreted it as the answer to the second part of question 2: "Is this a special case of a principle that any particle in the absence of external forces (excluding gravity) will always travel on a geodesic?" But because this answer would be unexpected for me, I googled and found a wikipedia page explaining Geodesics in general relativity. So what does the "No" really mean? –  Thomas Klimpel Aug 26 '13 at 7:47
    
I'm more reminded of "Ouroborus" in Red Dwarf than Marty in Back to the Future by your chicken and egg. –  WetSavannaAnimal aka Rod Vance Aug 26 '13 at 7:50
    
@ThomasKlimpel My answer had a lot of ambiguities. I edited it for greater clarity. I hope this version will avoid confusion. –  dj_mummy Aug 26 '13 at 13:55
    
@WetSavannaAnimalakaRodVance But Ouroborus is continuous. There were discontinuities in Marty's exit and entry at different events as far as onlookers were concerned. In the chicken's frame, Ouroborus would be more apt, but not in the frame of onlookers. –  dj_mummy Aug 26 '13 at 14:03
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